The Sutherland gauge

We here exhibit a symplectomorphism between the reduced phase space (Pred, Ωred) and the Sutherland phase space

M = T^∗C1 = C1× Rⁿ (2.46)

equipped with its canonical symplectic form, where C1 was defined in (2.2). As prepa-ration, we associate with any (q, p) ∈ M the G-element

Y (q, p) = K(q, p)− iκC, (2.47)

where K(q, p) is the 2n × 2n matrix

Kj,k =−Kn+j,n+k= ipjδj,k − µ(1 − δj,k)/ sin(qj − qk),

Kj,n+k =−K^n+j,k = (ν/ sin(2qj) + κ cot(2qj))δj,k+ µ(1− δ^j,k)/ sin(qj+ qk), (2.48) with j, k = 1, . . . , n. We also introduce the 2n-component vector

VR = (1, . . . , 1

| {z }

n times

,−1, . . . , −1| {z }

n times

)^⊤. (2.49)

Notice from (2.21) that K(q, p) ∈ G⁻.

Throughout the chapter we adopt the conditions (2.8) and take µ > 0, although the next result requires only that the real parameters µ, ν, κ satisfy

µ6= 0 and |ν| 6= |κ|. (2.50)

Theorem 2.1. Using the notations introduced in (2.22), (2.34) and (2.47), the subset S of the phase space P (2.26) given by

S =

(e^iQ(q), Y (q, p), υ^ℓ_µ,ν(VR), υ^r)| (q, p) ∈ M

, (2.51)

is a global cross-section for the action of G₊× G+ on P₀ = J⁻¹(0). Identifying P_red with S, the reduced symplectic form is equal to the Darboux form ω =Pn

k=1dqk∧ dp^k. Thus the obvious identification between S and M provides a symplectomorphism

(P_red, Ω_red)≃ (M, ω). (2.52)

Proof. We saw in Section2.2that the points of the level surface P₀ satisfy the equations

(yY y⁻¹)++ υ^ℓ_µ,ν(V ) = 02n and − Y+− iκC = 02n, (2.53) for some vector V ∈ C²ⁿ subject to CV + V = 0, V^†V = 2n. Remember that the block-form of any Lie algebra element Y ∈ G is

Y =

"

A B

−B^† D

#

with A + A^†= 0n= D + D^†, B ∈ C^n×n. (2.54)

Now the second constraint equation in (2.53) can be written as

2Y₊ =

"

A + D B− B^† B− B^† A + D

#

=

"

0_n −2iκ1n

−2iκ1n 0_n

#

=−2iκC, (2.55)

which implies that

D =−A and B^† = B + 2iκ1n. (2.56)

Thus every point of P0 has G-component Y of the form

Y =

"

A B

−B − 2iκ1n −A

#

with A + A^†= 0n, B ∈ C^n×n. (2.57)

By using the generalised Cartan decomposition (2.40) and applying a gauge transfor-mation (the action of G+ × G+ on P0), we may assume that y = e^iQ(q) with some q satisfying (2.38). Then the first equation of the momentum map constraint (2.53) yields the matrix equation

1

2i e^iQ(q)Y e^−iQ(q) + e^−iQ(q)CY Ce^iQ(q)

+ µ(V V^†− 12n) + (µ− ν)C = 02n. (2.58) If we introduce the notation V = (u, −u)^⊤, u ∈ Cⁿ, and assume that Y has the form (2.57) then (2.58) turns into the following equations for A and B

1

2i e^iqAe^−iq− e^−iqAe^iq

+ µ(uu^†− 1ⁿ) = 0n, (2.59)

and 1

2i e^iqBe^iq− e^−iqBe^−iq

− κe^−2iq− µuu^†+ (µ− ν)1ⁿ= 0n. (2.60) Since µ 6= 0, equation (2.59) implies that |u^j|² = 1 for all j = 1, . . . , n. Therefore we can apply a ‘residual’ gauge transformation by an element (gL, gR) = (e^iξ(x), e^iξ(x)), with suitable e^iξ(x) ∈ Z (2.37) to transform υ_µ,ν^ℓ (V ) into υ^ℓ_µ,ν(VR). This amounts to setting uj = 1 for all j = 1, . . . , n. After having done this, we return to equations (2.59) and (2.60). By writing out the equations entry-wise, we obtain that the diagonal components of A are arbitrary imaginary numbers (which we denote by ip₁, . . . , ip_n) and we also obtain the following system of equations

A_j,ksin(q_j − qk) =−µ = −Bj,ksin(q_j+ q_k), j 6= k,

Bj,jsin(2qj) = ν + κ cos(2qj)− iκ sin(2qj), j, k = 1, . . . , n. (2.61) So far we only knew that q satisfies π/2 ≥ q¹ ≥ · · · ≥ qⁿ ≥ 0. By virtue of the conditions (2.50), the system (2.61) can be solved if and only if π/2 > q1 >· · · > qn >

0. Substituting the unique solution for A and B back into (2.57) gives the formula Y = Y (q, p) as displayed in (2.47).

The above arguments show that every gauge orbit in P0 contains a point of S (2.51), and it is immediate by turning the equations backwards that every point of S belongs to P₀. By using that q satisfies strict inequalities and that all components of VR are

non-zero, it is also readily seen that no two different points of S are gauge equivalent.

Moreover, the effectively acting symmetry group, which is given by

(G+× G+)/U(1)diag (2.62)

where U(1) contains the scalar unitary matrices, acts freely on P0.

It follows from the above that Pred is a smooth manifold diffeomorphic to M. Now the proof is finished by direct computation of the pull-back of the symplectic form Ω of P (2.26) onto the global cross-section S.

Let us recall that the Abelian Poisson algebras Q¹ and Q² (2.32) consist of (G+× G+)-invariant functions on P , and thus descend to Abelian Poisson algebras on the re-duced phase space Pred. In terms of the model M ≃ S ≃ P^red, the Poisson algebra Q²_red is obviously generated by the functions (q, p) 7→ tr((−iY (q, p)))^m for m = 1, . . . , 2n.

It will be shown in the following section² that these functions vanish identically for the odd integers, and functionally independent generators of Q²_red are provided by the functions

Hk(q, p) = 1

4ktr(−iY (q, p))^2k, k = 1, . . . , n. (2.63) The first of these functions reads

H1(q, p) = 1 That is, upon the identification (2.7) it coincides with the Sutherland Hamiltonian (2.1). This implies the Liouville integrability of the Hamiltonian (2.1). Since its spec-tral invariants yield a commuting family of n independent functions in involution that include the Sutherland Hamiltonian, the Hermitian matrix function −iY (q, p) (2.47) serves as a Lax matrix for the Sutherland system (M, ω, H).

As for the reduced Abelian Poisson algebra Q¹_red, we notice that the cross-section S permits to identify it with the Abelian Poisson algebra of the smooth functions of the variables q₁, . . . , q_n. This is so since the level set P₀ lies completely in the ‘regular part’ of the phase space P , where the G-component y of (y, Y, υ^ℓ, υ^r) is such that Q(q) in its decomposition (2.40) satisfies strict inequalities π/2 > q1 > · · · > qⁿ > 0. It is a well-known fact that in the regular part the components of q are smooth (actually real-analytic) functions of y (while globally they are only continuous functions). To

2In fact, we shall see that Y (q, p) is conjugate to a diagonal matrix iΛ of the form in equation (2.71).

see that every smooth function depending on q ∈ C¹ is contained in Q¹_red, one may further use that every (G₊ × G+)-invariant smooth function on P0 can be extended to an invariant smooth function on P . Indeed, this holds since G₊× G+ is compact and P₀ ⊂ P is a regular submanifold, which itself follows from the free action property established in the course of the proof of Theorem 2.1.

We can summarize the outcome of the foregoing discussion as follows. Below, the generators of Poisson algebras are understood in the functional sense, i.e. if some f1, . . . , fn are generators then all smooth functions of them belong to the Poisson algebra.

Corollary 2.2. By using the model (M, ω) of the reduced phase space (Pred, Ωred) pro-vided by Theorem 2.1, the Abelian Poisson algebra Q²_red (2.31) can be identified with the Poisson algebra generated by the spectral invariants (2.62) of the ‘Sutherland Lax matrix’ −iY (q, p) (2.47), which according to (2.64) include the many-body Hamiltonian H(q, p) (2.1), and Q¹_red can be identified with the algebra generated by the corresponding position variables q_j (j = 1, . . . , n).